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Theorem normlem7 5069

Description: Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97.

**Hypotheses**
Ref	Expression
normlem1.1	⊢ S ∈ ℂ
normlem1.2	⊢ F ∈ ℋ
normlem1.3	⊢ G ∈ ℋ
normlem7.4	⊢ (abs ‘S) = 1

**Assertion**
Ref	Expression
normlem7	⊢ (((∗ ‘S) · (F ·_i G)) + (S · (G ·_i F))) ≤ (2 · ((√ ‘(G ·_i G)) · (√ ‘(F ·_i F))))

**Proof of Theorem normlem7**
Step	Hyp	Ref	Expression
1		normlem1.1	. . . . . 6 ⊢ S ∈ ℂ
2		normlem1.2	. . . . . 6 ⊢ F ∈ ℋ
3		normlem1.3	. . . . . 6 ⊢ G ∈ ℋ
4		cleqid 1102	. . . . . 6 ⊢ -(((∗ ‘S) · (F ·_i G)) + (S · (G ·_i F))) = -(((∗ ‘S) · (F ·_i G)) + (S · (G ·_i F)))
5	1, 2, 3, 4	normlem2 5064	. . . . 5 ⊢ -(((∗ ‘S) · (F ·_i G)) + (S · (G ·_i F))) ∈ ℝ
6	1	cjcl 4804	. . . . . . . 8 ⊢ (∗ ‘S) ∈ ℂ
7	2, 3	hicl 5044	. . . . . . . 8 ⊢ (F ·_i G) ∈ ℂ
8	6, 7	mulcl 4105	. . . . . . 7 ⊢ ((∗ ‘S) · (F ·_i G)) ∈ ℂ
9	3, 2	hicl 5044	. . . . . . . 8 ⊢ (G ·_i F) ∈ ℂ
10	1, 9	mulcl 4105	. . . . . . 7 ⊢ (S · (G ·_i F)) ∈ ℂ
11	8, 10	addcl 4104	. . . . . 6 ⊢ (((∗ ‘S) · (F ·_i G)) + (S · (G ·_i F))) ∈ ℂ
12	11	negre 4825	. . . . 5 ⊢ (-(((∗ ‘S) · (F ·_i G)) + (S · (G ·_i F))) ∈ ℝ ↔ (((∗ ‘S) · (F ·_i G)) + (S · (G ·_i F))) ∈ ℝ)
13	5, 12	mpbi 164	. . . 4 ⊢ (((∗ ‘S) · (F ·_i G)) + (S · (G ·_i F))) ∈ ℝ
14	13	leabs 4852	. . 3 ⊢ (((∗ ‘S) · (F ·_i G)) + (S · (G ·_i F))) ≤ (abs ‘(((∗ ‘S) · (F ·_i G)) + (S · (G ·_i F))))
15	11	absneg 4843	. . 3 ⊢ (abs ‘-(((∗ ‘S) · (F ·_i G)) + (S · (G ·_i F)))) = (abs ‘(((∗ ‘S) · (F ·_i G)) + (S · (G ·_i F))))
16	14, 15	breqtrr 2082	. 2 ⊢ (((∗ ‘S) · (F ·_i G)) + (S · (G ·_i F))) ≤ (abs ‘-(((∗ ‘S) · (F ·_i G)) + (S · (G ·_i F))))
17		cleqid 1102	. . 3 ⊢ (G ·_i G) = (G ·_i G)
18		cleqid 1102	. . 3 ⊢ (F ·_i F) = (F ·_i F)
19		normlem7.4	. . 3 ⊢ (abs ‘S) = 1
20	1, 2, 3, 4, 17, 18, 19	normlem6 5068	. 2 ⊢ (abs ‘-(((∗ ‘S) · (F ·_i G)) + (S · (G ·_i F)))) ≤ (2 · ((√ ‘(G ·_i G)) · (√ ‘(F ·_i F))))
21	11	negcl 4142	. . . 4 ⊢ -(((∗ ‘S) · (F ·_i G)) + (S · (G ·_i F))) ∈ ℂ
22	21	abscl 4840	. . 3 ⊢ (abs ‘-(((∗ ‘S) · (F ·_i G)) + (S · (G ·_i F)))) ∈ ℝ
23		2re 4470	. . . 4 ⊢ 2 ∈ ℝ
24		hiidge0t 5056	. . . . . . 7 ⊢ (G ∈ ℋ → 0 ≤ (G ·_i G))
25	3, 24	ax-mp 6	. . . . . 6 ⊢ 0 ≤ (G ·_i G)
26		hiidrclt 5053	. . . . . . . 8 ⊢ (G ∈ ℋ → (G ·_i G) ∈ ℝ)
27	3, 26	ax-mp 6	. . . . . . 7 ⊢ (G ·_i G) ∈ ℝ
28	27	sqrcl 4758	. . . . . 6 ⊢ (0 ≤ (G ·_i G) → (√ ‘(G ·_i G)) ∈ ℝ)
29	25, 28	ax-mp 6	. . . . 5 ⊢ (√ ‘(G ·_i G)) ∈ ℝ
30		hiidge0t 5056	. . . . . . 7 ⊢ (F ∈ ℋ → 0 ≤ (F ·_i F))
31	2, 30	ax-mp 6	. . . . . 6 ⊢ 0 ≤ (F ·_i F)
32		hiidrclt 5053	. . . . . . . 8 ⊢ (F ∈ ℋ → (F ·_i F) ∈ ℝ)
33	2, 32	ax-mp 6	. . . . . . 7 ⊢ (F ·_i F) ∈ ℝ
34	33	sqrcl 4758	. . . . . 6 ⊢ (0 ≤ (F ·_i F) → (√ ‘(F ·_i F)) ∈ ℝ)
35	31, 34	ax-mp 6	. . . . 5 ⊢ (√ ‘(F ·_i F)) ∈ ℝ
36	29, 35	remulcl 4119	. . . 4 ⊢ ((√ ‘(G ·_i G)) · (√ ‘(F ·_i F))) ∈ ℝ
37	23, 36	remulcl 4119	. . 3 ⊢ (2 · ((√ ‘(G ·_i G)) · (√ ‘(F ·_i F)))) ∈ ℝ
38	13, 22, 37	letr 4310	. 2 ⊢ (((((∗ ‘S) · (F ·_i G)) + (S · (G ·_i F))) ≤ (abs ‘-(((∗ ‘S) · (F ·_i G)) + (S · (G ·_i F)))) ∧ (abs ‘-(((∗ ‘S) · (F ·_i G)) + (S · (G ·_i F)))) ≤ (2 · ((√ ‘(G ·_i G)) · (√ ‘(F ·_i F))))) → (((∗ ‘S) · (F ·_i G)) + (S · (G ·_i F))) ≤ (2 · ((√ ‘(G ·_i G)) · (√ ‘(F ·_i F)))))
39	16, 20, 38	mp2an 520	1 ⊢ (((∗ ‘S) · (F ·_i G)) + (S · (G ·_i F))) ≤ (2 · ((√ ‘(G ·_i G)) · (√ ‘(F ·_i F))))

Colors of variables: wff set class

Syntax hints: = wceq 1091 ∈ wcel 1092 class class class wbr 2054 ‘cfv 2422 (class class class)co 3001 ℂcc 4026 ℝcr 4027 0cc0 4028 1c1 4029 + caddc 4031 · cmulc 4032 -cneg 4090 ≤ cle 4092 2c2 4454 √csqr 4727 ∗ccj 4788 abscabs 4789 ℋ chil 4958 ·_i csp 4963

This theorem is referenced by: normlem7t 5072 norm-ii 5086

This theorem was proved from axioms: ax-1 3 ax-2 4 ax-3 5 ax-mp 6 ax-4 673 ax-5 674 ax-6 675 ax-7 676 ax-gen 677 ax-8 798 ax-9 799 ax-10 800 ax-11 801 ax-12 802 ax-13 804 ax-14 805 ax-16 922 ax-17 925 ax-ext 1074 ax-rep 1075 ax-un 1076 ax-pow 1077 ax-reg 1078 ax-inf 1079 ax-hvaddcl 4984 ax-hvzercl 4987 ax-hvmulcl 4989 ax-hvmulass 4992 ax-hvmulzer 4995 ax-hicl 5043 ax-his1 5045 ax-his2 5046 ax-his3 5047 ax-his4 5048

This theorem depends on definitions: df-bi 128 df-or 197 df-an 198 df-3or 582 df-3an 583 df-ex 679 df-sb 853 df-eu 1009 df-mo 1010 df-clab 1093 df-cleq 1097 df-clel 1099 df-ne 1192 df-ral 1205 df-rex 1206 df-reu 1207 df-rab 1208 df-v 1349 df-sbc 1441 df-dif 1489 df-un 1490 df-in 1491 df-ss 1492 df-pss 1494 df-nul 1708 df-if 1777 df-pw 1799 df-sn 1811 df-pr 1812 df-tp 1814 df-op 1815 df-uni 1920 df-int 1966 df-iun 1996 df-tr 2042 df-br 2063 df-opab 2098 df-eprel 2122 df-id 2125 df-po 2128 df-so 2138 df-sup 2154 df-fr 2169 df-we 2186 df-ord 2202 df-on 2203 df-lim 2204 df-suc 2205 df-om 2373 df-xp 2424 df-rel 2425 df-cnv 2426 df-co 2427 df-dm 2428 df-rn 2429 df-res 2430 df-ima 2431 df-fun 2432 df-fn 2433 df-f 2434 df-f1 2435 df-fo 2436 df-f1o 2437 df-fv 2438 df-rdg 2970 df-opr 3003 df-oprab 3004 df-1st 3087 df-2nd 3088 df-1o 3104 df-oadd 3106 df-omul 3107 df-er 3200 df-ec 3202 df-qs 3205 df-ni 3794 df-pli 3795 df-mi 3796 df-lti 3797 df-plpq 3829 df-mpq 3830 df-enq 3831 df-nq 3832 df-plq 3833 df-mq 3834 df-rq 3835 df-ltq 3836 df-1q 3837 df-np 3880 df-1p 3881 df-plp 3882 df-mp 3883 df-ltp 3884 df-plpr 3958 df-mpr 3959 df-enr 3960 df-nr 3961 df-plr 3962 df-mr 3963 df-ltr 3964 df-0r 3965 df-1r 3966 df-m1r 3967 df-c 4034 df-0 4035 df-1 4036 df-i 4037 df-r 4038 df-plus 4039 df-mul 4040 df-lt 4041 df-sub 4133 df-neg 4135 df-div 4216 df-le 4277 df-n 4423 df-2 4462 df-3 4463 df-4 4464 df-n0 4535 df-z 4564 df-seq 4661 df-exp 4676 df-sqr 4728 df-re 4790 df-im 4791 df-cj 4792 df-abs 4793 df-hvsub 4996

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